NMSE Section 6.1 mentioned, B

Average Error: 14.2 → 9.2

Time: 21.1s

Precision: binary64

Cost: 7680

Math

\[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

↓

\[\left(\frac{\pi}{2} \cdot \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

FPCore

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))

↓

(FPCore (a b)
 :precision binary64
 (* (* (/ PI 2.0) (/ 1.0 (* (- b a) (+ b a)))) (- (/ 1.0 a) (/ 1.0 b))))

C

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

double code(double a, double b) {
	return ((((double) M_PI) / 2.0) * (1.0 / ((b - a) * (b + a)))) * ((1.0 / a) - (1.0 / b));
}

Java

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
}

↓

public static double code(double a, double b) {
	return ((Math.PI / 2.0) * (1.0 / ((b - a) * (b + a)))) * ((1.0 / a) - (1.0 / b));
}

Python

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b))

↓

def code(a, b):
	return ((math.pi / 2.0) * (1.0 / ((b - a) * (b + a)))) * ((1.0 / a) - (1.0 / b))

Julia

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b * b) - Float64(a * a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

↓

function code(a, b)
	return Float64(Float64(Float64(pi / 2.0) * Float64(1.0 / Float64(Float64(b - a) * Float64(b + a)))) * Float64(Float64(1.0 / a) - Float64(1.0 / b)))
end

MATLAB

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b * b) - (a * a)))) * ((1.0 / a) - (1.0 / b));
end

↓

function tmp = code(a, b)
	tmp = ((pi / 2.0) * (1.0 / ((b - a) * (b + a)))) * ((1.0 / a) - (1.0 / b));
end

Wolfram

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b * b), $MachinePrecision] - N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[a_, b_] := N[(N[(N[(Pi / 2.0), $MachinePrecision] * N[(1.0 / N[(N[(b - a), $MachinePrecision] * N[(b + a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] - N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 14.2

   \[\left(\frac{\pi}{2} \cdot \frac{1}{b \cdot b - a \cdot a}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

2. Applied egg-rr 9.2

   \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\color{blue}{\left(b - a\right) \cdot \left(b + a\right)}}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

3. Final simplification 9.2

   \[\leadsto \left(\frac{\pi}{2} \cdot \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\right) \cdot \left(\frac{1}{a} - \frac{1}{b}\right) \]

Alternatives

Alternative 1
Error	13.6
Cost	7824

\[\begin{array}{l} t_0 := \frac{1}{b \cdot b - a \cdot a}\\ t_1 := t_0 \cdot \left(0.5 \cdot \frac{\pi}{a}\right)\\ t_2 := -0.5 \cdot \frac{\pi}{b}\\ t_3 := \frac{1}{\left(b - a\right) \cdot \left(b + a\right)} \cdot t_2\\ \mathbf{if}\;b \leq -6.1 \cdot 10^{+159}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;b \leq -1.2 \cdot 10^{-38}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;b \leq 2.65 \cdot 10^{-15}:\\ \;\;\;\;t_0 \cdot t_2\\ \mathbf{elif}\;b \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_3\\ \end{array} \]

Alternative 2
Error	13.6
Cost	7688

\[\begin{array}{l} t_0 := \left(\frac{\pi}{2} \cdot \frac{1}{\left(b - a\right) \cdot \left(b + a\right)}\right) \cdot \frac{1}{a}\\ \mathbf{if}\;b \leq -2.8 \cdot 10^{-37}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq 5.4 \cdot 10^{-11}:\\ \;\;\;\;\frac{1}{b \cdot b - a \cdot a} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	23.1
Cost	7296

\[\frac{1}{\left(b - a\right) \cdot \left(b + a\right)} \cdot \left(-0.5 \cdot \frac{\pi}{b}\right) \]

Reproduce

herbie shell --seed 2023077 
(FPCore (a b)
  :name "NMSE Section 6.1 mentioned, B"
  :precision binary64
  (* (* (/ PI 2.0) (/ 1.0 (- (* b b) (* a a)))) (- (/ 1.0 a) (/ 1.0 b))))